Controllable mechanism of breathers in the (2+1)-dimensional nonlinear Schrödinger equation with different forms of distributed transverse diffraction

We study the (2+1)-dimensional nonlinear Schrödinger equation with different forms of distributed transverse diffraction in anisotropic graded-index grating waveguides, and obtain an exact two-breather solution for certain functional relations. From this solution, both Akhmediev breathers and Kuznetsov-Ma solitons can be constructed. A mechanism for controlling these localized solutions is presented. Two different transverse forms of diffraction and chirp factors play important roles in the evolutional characteristics such as phase, center and widths, while the gain/loss parameter only affects the evolution of their peaks. The propagation type of Akhmediev breathers and Kuznetsov-Ma solitons is determined by the relation between the maximum effective propagation distance, Zm, and the effective propagation distance, Z0, based on the center of the breathers. By adjusting this relation, partial excitation, maintenance and limitation of superposed Akhmediev breathers and Kuznetsov-Ma solitons are investigated for a waveguide with decreasing exponential diffraction.